Math 6420 • PDE

Let \(T:X\to X\) be a bounded linear operator defined on a Banach space \(X\). Denote by
\(R_\lambda\) the resolvent operator associated with \(T\) for some \(\lambda\in\rho(T)\).
The resolvent set of \(T\) is denoted by \(\rho(T)\).
\begin{enumerate}
\item[(a)] Use Neumann series to show that for \(\lambda,\mu\in\rho(T)\),
\[
  R_\lambda=\sum_{j=0}^{\infty}(\lambda-\mu)^j R_\mu^{j+1},
\]
where the series is absolutely convergent in the operator norm when
\[
  \lvert\lambda-\mu\rvert<\lVert R_\mu\rVert^{-1}.
\]

\item[(b)] Deduce from part (a) whether the spectrum \(\sigma(T)\) is open, closed or
neither.
\end{enumerate}